Analytical Solution of A Differential Equation that Predicts the Weather Condition by Lorenz Equations Using Homotopy Perturbation Method

Transcription

1 Globl Journl of Pur nd Applid Mhmis. ISSN Volum 3, Numbr 207, pp Rsrh Indi Publiions hp:// Anlyil Soluion of A Diffrnil Equion h Prdis h Whr Condiion by Lornz Equions Using Homoopy Prurbion Mhod S. Muhukumr, C. Thngpndi 2, S. Mhlkshmi 2, M. Vrmuni 2 Rgisrr,Tmil Univrsiy,Thnjvur, Tmilndu, Indi. 2 Dprmn of Mhmis, Th Mdur Collg, Mduri, Tmilndu, Indi. *Corrsponding uhor C : Thngpndi, Absr Th Lornz quion hs md qulifying hos possibl whih hs inspird mny mhmiins o rsrh nd sudy hos [2]. Chos hory is h brnh of mhmis fousd on h bhviour of dynmil sysms h r highly snsiiv o iniil sysms. Choi bhviour xiss in mny nurl sysms suh s whr nd lim. Th drminisi nur of h sysm dos no mk bhviour prdibl. This bhviour is known s drminisi hos or simply. Approxim nlyil soluion of Lornz quion is obind by Homoopy prurbion mhod HPM.Furhrmor, in his work numril simulion of h problm is lso rpord using Silb/Mlb progrm. Agrmns bwn nlyil nd numril rsuls r nod. Th nlyil rsul rpord in his work is usful o undrsnd h bhvior of h sysm. Kywords: Lornz quions; Chos; Homoopy prurbion mhod HPM; Mhmil modling; S vribls.. INTRODUCTION Th Lornz sysm is sysm of ordinry diffrnil quions firs sudid by Edwrd Lornz. In priulr, h Lornz ror is s of hoi soluions of h Lornz sysm whih whn plod rsmbl burfly or figur igh. Smll diffrns in iniil ondiions yild widly divrging ouoms for suh dynmil Corrsponding uhor:c. Thngpndimhs@gmil.om

2 8066 S. Muhukumr l sysms populrly rfrrd o s h burfly ff- rndring long -rm prdiion of h bhvior impossibl in gnrl. Th ph h ld o Lornz o hs quions bgn wih n ffor o find simpl modl problm whih h mhods usd for sisil whr forsing would fil. Th Lornz quions r lso onnd o ohr physil phnomnon []. Th Lornz quion hs md qulifying hos possibl whih hs inspird mny mhmiins o rsrh nd sudy hos [2]. Chos hory is h brnh of mhmis fousd on h bhvior of dynmil sysms h r highly snsiiv o iniil sysms. Choi bhvior xiss in mny nurl sysms suh s whr nd lim. Th drminisi nur of h sysm dos no mk bhvior prdibl. This bhvior is known s drminisi hos or simply hos. This bhvior n b sudid hrough nlysis of hoi mhmil modl. Chos hory hs ppliions in svrl disiplins suh s morology, soiology, physis, ology, onomis, biology. In hoi sysms, h unriny in fors inrss xponnilly wih h lpsd im. Th modl h inrodud [3] n b hough of gross simplifiion of on fur of mosphr nmly h fluid moion drivn by h hrml buoyny known s onvion. Th modl dsribs h onvion moion of fluid in smll, idlizd Ryligh Bnrd ll. Curry [4] hs shown h if h mod runion is no don, bu insd suffiin mods r rind o giv numril onvrgn, h hos dispprs. On h ohr hnd MLuglin nd Mrin [5] hs showd h hos is obind for hr dimnsion vrsion. Th xprimnl sysm dsribd by Lornz quions is h Riik dynmo homopolr gnror wih h oupu fd bk hrough induors nd rsisors o h oil gnring h mgni fild [6]. Th oupld irui nd roion quions n b rdud o h Lornz form nd xprimns [7]. In Mrh 963, Lornz wro h h wnd o inrodu, ordinry diffrnil quions whos soluion xmpl h simpls xmpl of drminisi non priodi flow nd fini mpliud onvion. In his ppr h xmins h work of Brry Slzmn nd John Ryligh whil inorporing svrl physil phnomn [8,9]. Lornz usd hr vribls o onsru simpl modl bsd on h 2 dimnsionl rprsnion of rh s mosphr. Th purpos of his ppr is o driv h pproxim xprssions of s vribls of Lornz quions using Homoopy prurbion mhod for ll vlus of prmrs. 2. Mhmil modling nd nlysis L us onsidr h diffrnil quion s follows dx x y d

3 Anlyil Soluion of A Diffrnil Equion h Prdis h Whr dy bx y xz 2 d dz d z xy 3 whr, b, nd r prmrs. Th iniil ondiions r A =0; x=, y=, z= 4 3. Anlyil soluion of Lornz quion using Homoopy Prurbion mhod Nonlinr sysm of quions plys n imporn rol in physis, hmisry nd biology. Consruing of priulr x soluion for hs quions rmins n imporn problm. In h ps mny uhors minly hd pid nion o sudy h soluion of nonlinr quions by using vrious mhods. Th Homoopy prurbion mhod hs bn workd ou ovr numbr of yrs by numrous uhors. Th Homoopy prurbion mhod HPM ws proposd by H nd ws sussfully pplid o uonomous ordinry diffrnil quions o nonlinr polyrysllin solids nd ohr filds. In his mhod h soluion produr is vry simpl nd only fw irions ld o high ur soluions whih r vlid for whol soluion domin. By solving h Eqns. - 4, using Homoopy Prurbion mhod w obin h nlyil soluions of Lornz quion s follows: x y - b b b 6

4 8068 S. Muhukumr l b z 7 4. DISCUSSION Eqns. 5-7 r h simpl nlyil xprssions of s vribl for ll vlus of prmrs, b nd. Figur, rprsns s vribl x vrsus im for fixd vlus of b=.2 nd =0.2. Figur 2, rprsns s vribl y vrsus im for fixd vlu of b=.5. Figur 3, rprsns s vribl z vrsus im for fixd vlus of =2 nd b = CONCLUSION Approxim nlyil soluions of h Lornz quions r prsnd using Homoopy Prurbion mhod. A simpl nd nw mhod of siming h s vribls r drivd. This soluion produr n b sily xndd o ll kinds of non-linr diffrnil quions wih vrious omplx boundry ondiions in nzym subsr rion diffusion prosss. Th xprssions providd in his work r usful o undrsnd h bhvior of h sysm. Rfrns [] Glik, J. Chos: Mking Nw Sin. Pnguin Books, Nw York, NY, 987. [2] S.H. Srogz, Nonlinr Dynmis nd hos. Addison Wsly, Rding, A, 994. [3] E. Lornz, Trnsions of h Nw York Admy of Sins, [4] J.H. Curry, Commun. Mh. Phys. 60, [5] J.B. MLughlin nd P.C. Mrin, Phys. Rv. A2, [6] E.A. Jkson, Cmbridg Univrsiy Prss, Vol.2, hp [7] K.A. Robbins, Mh. Pro. Cmb. Phil. So. 82, [8] Brdly, Lrry, Chos nd Frls [9] Viswnh, Diwkr. Th Frl Propry of h Lornz Aror. Physi D: Nonlinr Phnomn, Volum 90, Issus 2, Mrh 2004.

5 Anlyil Soluion of A Diffrnil Equion h Prdis h Whr Appndix A: Bsi onps of h Homoopy Prurbion mhod To xplin his mhod l us onsidr h following funion Aw fr 0; r A. Wih h boundry ondiions of w Bw, 0 ; r A.2 n Whr A, B, f r nd Г r gnrl diffrnil opror, boundry opror, known nlyi funion nd h boundry of h domin Ω rspivly. Gnrlly spking h opror A n b dividd ino linr pr L nd nonlinr pr N. Eqn. A. n b wrin s Lw Nw fr 0 A.3 W onsru homoopy zr,p : Ω x [0,] R whih sisfis Hz,p p[ L w L z] p[ A z f r] 0, p [0,],r 0 or Hz,p Lw0 L z pl w0 p[ N z f r] 0 A.4 whr p ϵ [0,] is n mbdding prmr, whil w0 is n iniil pproximion of Eqn.A.,whih sisfis h boundry ondiions. Obviously, from Eqn.A.4 w will hv A.5 Hz,0 Lz Lw0 0 Hz, A z f r A.6 Th hnging pross of p from zro o uniy is jus h of zr, p from w0 o wr.in Topology, his is lld dformion, whil L z L w nd A z f r r lld 0 Homoopy. Aording o h HPM, w n firs us mbdding prmr p s smll prmr, nd ssum h h soluions of Eqns. A.3 nd A.4 n b wrin s powr sris in p. 2 z z pz p z... A sing p= rsuls in h pproxim soluion of Eqn.A. w lim p z z0 z z2... A.8 Th ombinion of h Prurbion mhod nd h Homoopy mhod is lld h HPM, whih limins h drwbks of h rdiionl Prurbion mhods whil kping ll is dvngs.

6 8070 S. Muhukumr l Appndix B: Anlyil soluions for h dimnsionlss onnrions In his ppndix, w indi how Eqns. 5 7 in his ppr hs bn drivd. To find h soluions of Eqn., w onsru Homoopy s follows, from Eqn.A., dx dx p x p x y 0 B. d d Th iniil pproximions r s follows 2 x x0 px p x... B dx0 P : x0 0 d B.3 dx P : x y0 0 d B.4 dx P 2 2 : x2 y 0 d B.5 Solving h bov Eqns. B. o B.5, w g x 0 B.6 x B.7 b x2 Adding h bov Eqns. B.6 o B.8, w g Eqn. 5 Similrly w n obin Eqns. 6 nd 7. Appndix C funion p4num opions= ods 'RlTol',-6,'Ss','on'; %iniil ondiions Xo= [; ; ]; spn = [0,]; i [,X] = od45@tsfunion,spn,xo,opions; o B.8

7 Anlyil Soluion of A Diffrnil Equion h Prdis h Whr. 807 figur hold on %plo, X:,,'-' plo, X:,2,'-' %plo, X:,3,'-' lgnd'x','x2','x3' ylbl'x' xlbl'' rurn funion [dx_d]= TsFunion,x =2;b=0.0;=; dx_d = -*x+*x2; dx_d2 =b*x-x2-x*x3; dx_d3 =-*x3+x*x2; dx_d = dx_d'; rurn Figur : Plo of S vribl x vrsus im vrious vlus of h prmrs. Solid lins rprsn numril soluions whrs h dod lin rprsns nlyil soluions.

8 8072 S. Muhukumr l Figur 2: Plo of S vribls x vrsus im vrious vlus of h prmrs. Solid lins rprsn numril soluions whrs h dod lin rprsns nlyil soluions. Figur 3: Plo of S vribls y vrsus im vrious vlus of h prmrs nd. Solid lins rprsn numril soluions whrs h dod lin rprsns nlyil soluions.

9 Anlyil Soluion of A Diffrnil Equion h Prdis h Whr Figur 4: Plo of S vribls y vrsus im vrious vlus of h prmrs nd. Solid lins rprsn numril soluions whrs h dod lin rprsns nlyil soluions. Figur 5: Plo of S vribls z vrsus im vrious vlus of h prmrs nd b.

10 8074 S. Muhukumr l Solid lins rprsn numril soluions whrs h dod lin rprsns nlyil soluions. Figur 6: Plo of S vribls z vrsus im vrious vlus of h prmrs nd b. Solid lins rprsn numril soluions whrs h dod lin rprsns nlyil soluions.